17.2. HAHN BANACH THEOREM 443

Therefore, if dyndx → f ∈C ([0,1]) and if yn→ y in C ([0,1]) it follows that

yn (x) =∫ x

0dyn(t)

dx dt↓ ↓

y(x) =∫ x

0 f (t)dt

and so by the fundamental theorem of calculus f (x) = y′ (x) and so the mapping is closed.It is obviously not continuous because it takes y(x) and y(x)+ 1

n sin(nx) to two functionswhich are far from each other even though these two functions are very close in C ([0,1]).Furthermore, it is not defined on the whole space, C ([0,1]).

The next theorem, the closed graph theorem, gives conditions under which closed im-plies continuous.

Theorem 17.1.17 Let X and Y be Banach spaces and suppose L : X → Y is closed andlinear. Then L is continuous.

Proof: Let G be the graph of L. G = {(x,Lx) : x ∈ X}. By Lemma 17.1.15 it followsthat G is a Banach space. Define P : G→ X by P(x,Lx) = x. P maps the Banach space Gonto the Banach space X and is continuous and linear. By the open mapping theorem, Pmaps open sets onto open sets. Since P is also one to one, this says that P−1 is continuous.Thus ||P−1x|| ≤ K||x||. Hence

||Lx|| ≤max(||x||, ||Lx||)≤ K||x||

By Theorem 17.1.4 on Page 437, this shows L is continuous and proves the theorem.The following corollary is quite useful. It shows how to obtain a new norm on the

domain of a closed operator such that the domain with this new norm becomes a Banachspace.

Corollary 17.1.18 Let L : D⊆ X → Y where X ,Y are a Banach spaces, and L is a closedoperator. Then define a new norm on D by

||x||D ≡ ||x||X + ||Lx||Y .

Then D with this new norm is a Banach space.

Proof: If {xn} is a Cauchy sequence in D with this new norm, it follows both {xn} and{Lxn} are Cauchy sequences and therefore, they converge. Since L is closed, xn → x andLxn→ Lx for some x ∈ D. Thus ||xn− x||D→ 0.

17.2 Hahn Banach TheoremThe closed graph, open mapping, and uniform boundedness theorems are the three majortopological theorems in functional analysis. The other major theorem is the Hahn-Banachtheorem which has nothing to do with topology. Before presenting this theorem, here aresome preliminaries about partially ordered sets.